Yes. Hourtick, TimeCamp, timr, Clockodo and solidtime are free or have a free plan, according to the pricing each company publishes. Free plans usually carry limits; each product page links to the company's terms.
Which European time tracking tools are open source?
Kimai and solidtime publish their source code under an open-source licence. Each product page shows the licence and where the code is.
Which European time tracking tools can you host yourself?
Kimai and solidtime can be run on your own servers, according to their own documentation.
